Many prior art on optimizing heuristic function [1]}, [2]}, [3]}, [4]}, [5]}
minimize the \(\mathrm {L}_2\)  loss functionWhile some works use \(L_1\) , the properties discussed here for \(L_1\)  holds for \(\mathrm {L}_2\)  as well. \(\mathrm {L}_2(h_\theta , \mathcal {S}) = \sum _{i} \left(h_\theta (s_i) - y_i\right)^2,\)  where the training set \(\mathcal {S}\)  consists of pairs \(\left\lbrace (s_i, y_i)\right\rbrace _{i=1}^n,\)  where \(s_i\)  is some state and \(y_i\)  is the length of the plan from \(s_i\)  to the goal state. We argue that zero \(\mathrm {L}_2\)  loss on a given problem instance for states on the optimal path does not guarantee that A* will be optimally efficient in the sense that it can expand more states than needed.
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